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On orthogonality sampling method for Maxwell's equations and its applications to experimental data

Thu Le, Dinh-Liem Nguyen

TL;DR

The paper tackles the inverse scattering problem for Maxwell's equations in a bounded bi-anisotropic medium at a fixed frequency, aiming to reconstruct the location and shape of the scatterer from multi-static far-field data. It develops a factorization analysis showing uniqueness and uses it to justify a Modified Orthogonality Sampling Method (MOSM) extended to bi-anisotropic media, linking the imaging functional to the far-field operator via $\mathcal{F}=\mathcal{H}^{*}\mathcal{T}\mathcal{H}$. The authors prove a resolution bound and noise-stability for MOSM, and validate the approach through synthetic experiments and real, unprocessed Fresnel Institute data, demonstrating robustness to high noise and limited aperture data. The work offers a fast, non-iterative inversion tool with practical potential for engineering applications in non-destructive testing and radar imaging where bi-anisotropic effects are significant.

Abstract

This paper addresses the inverse scattering problem for Maxwell's equations. We first show that a bianisotropic scatterer can be uniquely determined from multi-static far-field data through the factorization analysis of the far-field operator. Next, we investigate a modified version of the orthogonality sampling method, as proposed in Le [2022 Inverse Problems 38 025007], for the numerical reconstruction of the scatterer. Finally, we apply this sampling method to invert unprocessed 3D experimental data obtained from the Fresnel Institute. Numerical examples with synthetic scattering data for bianisotropic targets are also presented to demonstrate the effectiveness of the method.

On orthogonality sampling method for Maxwell's equations and its applications to experimental data

TL;DR

The paper tackles the inverse scattering problem for Maxwell's equations in a bounded bi-anisotropic medium at a fixed frequency, aiming to reconstruct the location and shape of the scatterer from multi-static far-field data. It develops a factorization analysis showing uniqueness and uses it to justify a Modified Orthogonality Sampling Method (MOSM) extended to bi-anisotropic media, linking the imaging functional to the far-field operator via . The authors prove a resolution bound and noise-stability for MOSM, and validate the approach through synthetic experiments and real, unprocessed Fresnel Institute data, demonstrating robustness to high noise and limited aperture data. The work offers a fast, non-iterative inversion tool with practical potential for engineering applications in non-destructive testing and radar imaging where bi-anisotropic effects are significant.

Abstract

This paper addresses the inverse scattering problem for Maxwell's equations. We first show that a bianisotropic scatterer can be uniquely determined from multi-static far-field data through the factorization analysis of the far-field operator. Next, we investigate a modified version of the orthogonality sampling method, as proposed in Le [2022 Inverse Problems 38 025007], for the numerical reconstruction of the scatterer. Finally, we apply this sampling method to invert unprocessed 3D experimental data obtained from the Fresnel Institute. Numerical examples with synthetic scattering data for bianisotropic targets are also presented to demonstrate the effectiveness of the method.

Paper Structure

This paper contains 8 sections, 6 theorems, 63 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Unique determination and factorization analysis
  4. Modified orthogonality sampling method
  5. Numerical study
  6. Synthetic data
  7. Raw experimental data
  8. Conclusion

Key Result

Lemma 1

Suppose that Assumption I holds, then the middle operator $\mathcal{T}$ defined in defT is linear and bounded on $[L^2(\Omega)]^3\times [L^2(\Omega)]^3$.

Figures (6)

  • Figure 1: A diagram illustrating the geometry of the scattering problem.
  • Figure 2: Reconstruction results of an L-shaped scatterer with synthetic data at $k=12$: (a) true scatterer; (b) reconstruction with $30\%$ noise; (c) reconstruction with $50\%$ noise; (d)--(f) 2D cross-sectional views of (a)--(c) on $\{y=0\}$, respectively.
  • Figure 3: MOSM, FM and OSM reconstruction of TwoSpheres using raw experimental data at $4$ GHz.
  • Figure 4: MOSM reconstruction of TwoCubes using raw experimental data at $7.5$ GHz.
  • Figure 5: MOSM reconstruction of CubeSpheres using raw experimental data at $4.75$ GHz.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 1 more